Step by step
The question
Question: solve tan x + 1 = 0.
Find every angle from 0° up to 360°.
We want the angles from 0° to 360°: one full turn.
1 Remove the + 1
We want tan x by itself.
The + 1 is in the way.
To remove + 1, we take away 1.
We do it on both sides.
Why both sides?
The = sign means both sides are the same amount.
If we change only one side, they are not the same any more.
+ 1 and − 1 make zero.
Cross them out.
0 − 1 = −1.
So tan x = −1!
2 Find the reference angle
From the unit circle: tan 45° = 1.
So the reference angle is 45°.
The minus only changes which quadrants.
3 Find every angle
Tan is negative in quadrants II and IV.
180° − 45° and 360° − 45°: x = 135° or 315°.
Careful!
The calculator gives only 135°.
But the line crosses the circle twice.
315° works too.
On the circle: 2 points, so 2 angles.
So x = 135°, 315°.
4 Check
Let's check every answer in the equation.
Each one must give 0.
Every answer works.
So x = 135°, 315°.
Let's look back.
First: remove the + 1.
Then: find the reference angle.
Then: find every angle.
Last: check.